L(s) = 1 | + (0.324 − 0.945i)2-s + (−0.986 + 0.164i)3-s + (−0.789 − 0.614i)4-s + (0.879 + 0.475i)5-s + (−0.164 + 0.986i)6-s + (−0.915 − 0.401i)7-s + (−0.837 + 0.546i)8-s + (0.945 − 0.324i)9-s + (0.735 − 0.677i)10-s + (0.0825 − 0.996i)11-s + (0.879 + 0.475i)12-s + (−0.475 + 0.879i)13-s + (−0.677 + 0.735i)14-s + (−0.945 − 0.324i)15-s + (0.245 + 0.969i)16-s + (−0.879 − 0.475i)17-s + ⋯ |
L(s) = 1 | + (0.324 − 0.945i)2-s + (−0.986 + 0.164i)3-s + (−0.789 − 0.614i)4-s + (0.879 + 0.475i)5-s + (−0.164 + 0.986i)6-s + (−0.915 − 0.401i)7-s + (−0.837 + 0.546i)8-s + (0.945 − 0.324i)9-s + (0.735 − 0.677i)10-s + (0.0825 − 0.996i)11-s + (0.879 + 0.475i)12-s + (−0.475 + 0.879i)13-s + (−0.677 + 0.735i)14-s + (−0.945 − 0.324i)15-s + (0.245 + 0.969i)16-s + (−0.879 − 0.475i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.086424542 - 0.03162034083i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.086424542 - 0.03162034083i\) |
\(L(1)\) |
\(\approx\) |
\(0.7959878735 - 0.2827342660i\) |
\(L(1)\) |
\(\approx\) |
\(0.7959878735 - 0.2827342660i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 229 | \( 1 \) |
good | 2 | \( 1 + (0.324 - 0.945i)T \) |
| 3 | \( 1 + (-0.986 + 0.164i)T \) |
| 5 | \( 1 + (0.879 + 0.475i)T \) |
| 7 | \( 1 + (-0.915 - 0.401i)T \) |
| 11 | \( 1 + (0.0825 - 0.996i)T \) |
| 13 | \( 1 + (-0.475 + 0.879i)T \) |
| 17 | \( 1 + (-0.879 - 0.475i)T \) |
| 19 | \( 1 + (-0.879 + 0.475i)T \) |
| 23 | \( 1 + (0.735 + 0.677i)T \) |
| 29 | \( 1 + (0.915 + 0.401i)T \) |
| 31 | \( 1 + (0.996 + 0.0825i)T \) |
| 37 | \( 1 + (0.245 - 0.969i)T \) |
| 41 | \( 1 + (-0.324 + 0.945i)T \) |
| 43 | \( 1 + (0.245 - 0.969i)T \) |
| 47 | \( 1 + (0.324 + 0.945i)T \) |
| 53 | \( 1 + (-0.986 + 0.164i)T \) |
| 59 | \( 1 + (0.969 - 0.245i)T \) |
| 61 | \( 1 + (0.945 - 0.324i)T \) |
| 67 | \( 1 + (0.324 + 0.945i)T \) |
| 71 | \( 1 + (0.0825 + 0.996i)T \) |
| 73 | \( 1 + (0.837 - 0.546i)T \) |
| 79 | \( 1 + (0.915 - 0.401i)T \) |
| 83 | \( 1 + (0.245 + 0.969i)T \) |
| 89 | \( 1 - iT \) |
| 97 | \( 1 + (-0.546 + 0.837i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−25.795773048405265408113305427141, −25.09638950916356930520361308312, −24.412997753437167435544265647098, −23.31836303714018821745013953477, −22.47478370852127858259424865021, −21.97930091943721006051059818241, −20.91827117421372217013284865920, −19.3900948888659871561471317302, −18.12356218832977994310345935562, −17.39312988514078551099358643642, −16.89604708138910576569621188283, −15.71053934406173973927315993393, −15.04318671864093059323659251287, −13.41815435854838911137075630223, −12.80953027223875070911102879119, −12.20493343559026342404095095312, −10.363679264146155542877408425615, −9.55237441242995642263082538623, −8.36169871791076846962337659933, −6.767727149058991568533493557144, −6.35471011435505049845183952, −5.19625287735897921952347552761, −4.447778584971324925676036760590, −2.47089928087530322723042926960, −0.46461730498994548261281192738,
0.9506148338464927493213372287, 2.46067017965072627140329543605, 3.74870808243042332591350980424, 4.94306193014106329760005369330, 6.1264074532866542105079097927, 6.75311231804302075162386718117, 9.04942051033023440756031633126, 9.86678413729683593903348978413, 10.7181335496201030635653273521, 11.46152243315879175407600562058, 12.66117437288635891661047121240, 13.47172641814503774018819812460, 14.33518702179313899004828499396, 15.779717793877570647424596367261, 16.92669679784899695065855944300, 17.67528216758089601013391585413, 18.83888013737421068161072885715, 19.35833675663603858283176884779, 20.88768062986052764787496723842, 21.687659032953283115450307463703, 22.17633690573720603571679802743, 23.09739612023082452254677936362, 23.87350437783755932425811689160, 25.119584170409691381266574558056, 26.6918260754485552310625872624